Vibration of Periodic Structures

By Gautam SenGupta

Vibration of Periodic Structures introduces the fundamentals of periodic structure theory by considering the simplest model – wave propagation in an infinitely long periodic spring-mass system. It then shows how the knowledge of the stop and pass bands can be utilized to find the natural frequency distribution in a finite periodic structure. The basic concepts are further extended to wave propagation in infinitely long periodically supported beams and plates; distribution of natural frequencies of a similar structure of finite length; vibration of skin-stringer structures; and structuralacoustic properties of a section of an aircraft fuselage, based on a combination of the finite element method and the periodic structure theory, in a highly cost-effective manner.

This book is a valuable resource of information for practicing engineers in various industries, e.g., civil, mechanical, or aerospace engineering, dealing with vibration of structures with periodic properties, including prediction of supersonic flutter characteristics of aerospace structures. It will also prove to be a beneficial reference for researchers involved with wave propagation in metamaterials and phononic devices.

“Readers who have wanted a clear and connected account of vibration of periodic structures will find this treatment accessible and stimulating and will want to add this volume to their personal or institutional library. – Prof. Earl Dowell, Duke University, Durham, NC, USA

• Shows how the periodic structure theory can be combined with the finite element method to model a section of an airplane fuselage to study its structural-acoustic characteristics
• Features developing methods for predicting the dynamics of periodic structures in a cost-effective manner
• Guides the reader to predict and reduce response of periodically stiffened structures to random excitations

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 27, 2023
• ISBN: 9780323990233

Gautam SenGupta

Dr. Gautam SenGupta received his BSc (Physics) from Presidency College, Calcutta, India, his BTech (Mechanical Engineering) from IIT, Kharagpur, India, and his PhD from the University of Southampton, United Kingdom, where he developed a unique method for predicting the natural frequencies of periodic structures. He worked as a Specialist Engineer in Acoustic Fatigue at the Royal Aeronautical Society of England, and at NASA-Langley as a Research Associate. He joined Boeing in 1973, where he applied the Periodic Structure Theory to predict and control the structural acoustic characteristics of an airplane fuselage in a cost-effective manner. He retired in 2016 as a Technical Fellow of the Boeing Company in Seattle, where he was also involved in developing methods for predicting airframe noise and transonic flutter characteristics of airplanes. In addition, he has served as an Affiliate Professor at the University of Washington, where he taught courses on Structural Dynamics and related subjects.

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Preface

Before delving into the subject of this book, it is interesting to note that wave propagation in a periodic spring–mass system has its origin in the work of Newton [1]. He used such a model to determine the speed of sound in air, assuming sound propagation in air was similar to propagation of elastic waves along an infinite spring–mass system. Further historic details on earlier work by Newton, Laplace, John and Daniel Bernoulli, Euler, Kelvin, and other stalwarts of science can be seen in the classic book by Brillouin [2]. Vibration of a uniform string was also analyzed by Rayleigh [3] by replacing it with a periodic distribution of concentrated masses along its length. A comprehensive summary of historical origins and the future outlook for the periodic structure theory in the context of phononic materials and structures can be found in [4].

When I arrived at Southampton in October 1967 to start my graduate studies under the guidance of Professor D.J. Mead, I had no idea of what I was getting into. As described by Mead [5], Periodic structures were first studied at the University of Southampton in 1964 in the context of noise excited vibration and sonic fatigue of stiffened aerospace structures. Large areas of these consist of uniform plates and shells with identical stiffeners at regular intervals, and research into their natural frequencies, modes and random response levels was required with a view to predicting stress levels and fatigue endurance. This feeling was subsequently justified by our own calculations and also by those of Lin [6], Lust [7] and others. As we proceeded, we found that periodic structure theory was well suited to lightly damped as well as to heavily damped finite periodic structures.

As I became familiar with Mead's work based on the theory of wave propagation in infinite periodic structures, work done by Lin [6] and Miles [8] based on difference equations and the transfer matrix method, it gradually became clear to me that these different methods were complementary to each other from the mathematical perspective. However, while the three methods accurately predicted the response of periodic structures to random excitations, they did not quite address the physics hidden behind pages and pages of mathematics. They also did not answer why the natural frequencies appeared in groups, with the number of frequencies in each group being exactly equal to the number of spans of the finite periodic structures, and why, for an infinite periodic structure, the cluster of discrete frequencies in each group of a finite periodic structure blended into continuous frequency bands in which the flexural waves could propagate freely.

My curiosity led me to investigate the issue by focusing on the physics of the problem, which gave rise to a clear explanation of the clustering of the frequencies in each propagation band. The work showed that at the natural frequencies, only certain discrete values of the wave propagation constant satisfy the boundary conditions, and an integral multiple of half or quarter (depending on the boundary conditions) of the primary structural wavelength fits into the length of the structure. This led me to develop a simple and convenient graphical method for finding the natural frequency distribution for different boundary conditions of a finite periodic structure [9,10]. The method has subsequently been applied to prediction of natural frequencies of orthogonally stiffened periodic plates and shells, periodic model of a section of an airplane fuselage, and, to my pleasant surprise, to prediction of supersonic flutter characteristics of periodically supported plates and shells [11,12], and many other applications, e.g., design of the Brabau bridge in Italy [13], discussed in Chapter 14.

Looking back, it is gratifying to see that some of our early research done at Southampton has also benefited the phononics community [4,14–16].

Subsequently, I was able to continue my research on vibration of periodic structures at NASA-Langley and Boeing, and apply some of the ideas to analyze sections of an aircraft fuselage modeled as a periodic structure, based on a combination of the finite element method with the periodic structure theory in a cost-effective manner, and development of innovative damping concepts for potential reduction of structural-acoustic response of an airplane fuselage excited by jet noise and boundary layer turbulence.

While at Boeing, I was also involved in teaching short courses on related subjects to Boeing engineers as well as others through the American Institute of Aeronautics (AIAA). Since periodic structure theory is not commonly taught in engineering schools, I decided to put together this book based on the courses I taught and the vast literature available in various journals and conference papers. I hope future civil, mechanical, aerospace, and phononics engineers will find this book useful in the course of their work, and professors in the above disciplines will find the book useful for developing courses suitable to their needs.

Choosing what to cover in this book, and in what order, was not easy. After much deliberation, I decided to start with the simplest example of a periodic system that would be easy to follow for beginners starting their journey, regardless of which engineering discipline they were coming from, and gradually moved up to systems with added complexities so that the models considered began to look like real structures one encounters in engineering practice. In the process, I sincerely regret that discussions of many other important contributions could not be included in this edition. Hopefully, the interested and motivated readers will be able to learn about those, and other valuable contributions during the course of their work.

In order to make the book suitable for teaching a senior undergraduate or graduate level course in civil, mechanical, aeronautical, and phononics engineering, the chapters are introduced in the following sequence:

Chapter 1 deals with the fundamentals of the periodic structure theory, where the basic ideas are introduced by looking at the simplest possible examples, i.e., an infinitely long periodic spring–mass system and then looking at the distribution of the natural frequencies in a finite, periodic spring–mass system with different boundary conditions, using conventional approach, and the approach based on the periodic structure theory.

Chapter 2 introduces the readers to continuous structural elements such as beams and plates, wave propagation in such elements, the natural frequencies and normal modes for different boundary conditions, and the concept of coincidence excitation in infinite beams and plates.

Chapter 3 covers the subject of propagation of flexural waves in periodically supported infinite beams and plates, introduces the concept of pass and stop bands, and the mechanism of coincidence excitation by an incident acoustic wave or a convected pressure field.

Chapter 4 goes into explaining how the natural frequencies of finite periodically supported beams and plates with various boundary conditions, and why the conditions at the extreme ends of a finite periodic structure permit only certain discrete values of the propagation constant. It also provides a better physical understanding of the phenomenon involved.

Chapter 5 shows how the basic concepts introduced in the previous chapter can be extended to deals with periodically stiffened panels such as skin–stringer structures of finite length representing a real structure such as a section of an airplane fuselage, a bridge with identical sections, or a tall building.

Chapter 6 discusses wave propagation in doubly periodic structures, consisting of the repetition of a basic periodic unit, which is a periodic structure in itself. It is seen that the propagation zones in these structures are distributed in a pattern which is to some extent doubly periodic in appearance. The bounding frequencies of the propagation zones can be identified with the natural frequencies of the multi-span periodic structure constituting the basic unit with various end conditions.

Chapter 7 deals with response of, and sound radiation from periodic structures, based on the concept of space-harmonics, and the basic concepts are illustrated with simple examples.

Chapter 8 shows how to combine the finite element method (FEM) with the periodic structure (PS) theory for modeling the complexities of real periodic structures in a cost effective manner. The basic ideas are illustrated with simple examples, e.g., a periodically supported beam. Applications to acoustics are also discussed with examples of acoustic modes of rigid pipes with various boundary conditions, and a toroidal cavity.

In Chapter 9, a method is presented for predicting the natural modes and frequencies of periodic structures coupled with an enclosed fluid medium. In this method, the structure and the fluid medium of the basic periodic unit are first modeled using the finite element method (FEM). The periodic structures (PS) theory is then applied to the mass and stiffness matrices of the basic periodic unit extracted from the finite element program, to account for the periodicity of the structure. By using the combined FEM–PS method, natural modes of the entire coupled structure–fluid system can be computed from the resulting matrix difference equation. It is shown that significant cost saving can be achieved, both in terms of computer time and memory, by using the proposed FEM–PS method. This method can be particularly effective early in design, e.g., for identifying design parameters that are important for reducing the interior noise in an aircraft cabin. It is a pleasure to acknowledge the support received from Mr. Hamid Jamshidiat (a colleague at Boeing) during the course of the work presented in this